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Unformatted text preview: 1 INTRODUCTION Lecture 1 ORIE3500/5500 Summer2009 Sets and Probabilities 1 Introduction What is probability? The following examples will throw some light. Example A coin is tossed a large number of times, like a million and suppose that 40% of the times it tosses Heads. We shall say that in a single toss, the probability of Heads is 0 . 4. Example What does it mean to say “chances of rain tomorrow is 20%”? An answer: If we look at a large number of similar (where the prediction of rain is 20%) days, then the proportion of days in which it rained is close to . 2. Example What does the statement “That an underage (below 16) driver will have an accident is twice as likely as the same for an adult driver” mean? Proportion of underage drivers who had an accident = 2 Proportion of underage drivers who had an accident (and that’s why they are not allowed to drive!) An intuitive definition : Probability of an event can be defined as the proportion of times it will occur if the experiment is repeated independently under identical setup a large number of times. In order to understand the above definition, need clear notions about the following: • event • repeat independently • identical setup Shall make these notions precise as we go along in the course. 1 2 SETS 2 Sets Probability makes extensive use of set operations, so let us introduce at the outset the relevant notations and terminology. Definition. A set is a collection of objects. The objects are elements of the set. Notations : • If S is a set and x is an element of S , we write x ∈ S . • If S is a set and x is NOT an element of S , we write x / ∈ S . • A set can have no elements, in which case it is called the empty set, and is denoted by φ . Definition. If every element of a set S is also an element of a set T , we say S is a subset of T and write S ⊂ T (or equivalently, T is a supset of S , written as T ⊃ S )....
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.
- Summer '08